Principles of
Managerial Finance
Time Value of Money
MBA 656 FINANCIAL MANAGEMENT CYCLE 1
BY: MARY ROSE HABAGAT
GELITA COL...
WHY THIS TOPIC MATTERS
TO YOU
IN PROFESSIONAL LIFE:
ACCOUNTING:
You need to understand time-value-of-money
calculations to...
INFORMATION SYSTEM:
You need to understand time-value-of-money
calculations to design systems that accurately
measure and ...
MARKETING
You need to understand time value of money
because funding for new programs and products
must be justified finan...
IN YOUR PERSONAL LIFE
Time value techniques are widely used in
personal financial planning. You can use them
to calculate ...
Learning Objectives
• Discuss the role of time value in finance and the use
of computational aids used to simplify its app...
Learning Objectives
• Calculate the present value of a mixed stream of cash
flows, an annuity, a mixed stream with an embe...
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Tim...
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Tim...
Present Value and Future Value
PRESENT VALUE

FUTURE VALUE

•

Is the cash on hand today

•

•

It is the amount you need
...
ILLUSTRATION
Simple Interest
With simple interest, you don’t earn interest on
interest.
• Year 1: 5% of $100 =

$5 + $100 = $105

• Yea...
Compound Interest
With compound interest, a depositor earns interest
on interest!
• Year 1: 5% of $100.00 = $5.00 + $100.0...
Computational Aids

• Use the Equations
• Use the Financial Tables
• Use Financial Calculators
• Use Spreadsheets
Computational Aids

Future value interest factor or present value interest factor
Computational Aids
Time Value Terms
• PV0

=

present value or beginning amount

• k

=

interest rate

• FVn

=

future value at end of “n” ...
Four Basic Models
• FVn

=

PV0(1+k)n

=

PV(FVIFk,n)

• PV0

=

FVn[1/(1+k)n]

=

FV(PVIFk,n)

A (1+k)n - 1
k

=

A(FVIFA...
BASIC PATTERNS OF CASH FLOW
• SINGLE AMOUNT: a lump sum amount
either currently held or expected at some
future date
• ANN...
Future Value Example
Algebraically and Using FVIF Tables
You deposit $2,000 today at 6%
interest. How much will you have i...
Future Value Example
Using Excel
You deposit $2,000 today at 6%
interest. How much will you have in 5
years?

PV
k
n
FV?

...
Compounding More Frequently
than Annually
• Compounding more frequently than once a year
results in a higher effective int...
Compounding More Frequently
than Annually
• For example, what would be the difference in future
value if I deposit $100 fo...
Compounding More Frequently
than Annually
On Excel
Annually
PV

$

Sem iAnnually Quarterly

100.00

k

12.0%

n

5

FV

$1...
Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Thr...
Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Thr...
Present Value Example
Algebraically and Using PVIF Tables
How much must you deposit today in order to
have $2,000 in 5 yea...
Present Value Example
Using Excel
How much must you deposit today in order to
have $2,000 in 5 years if you can earn 6%
in...
Annuities
• Annuities are equally-spaced cash flows of equal size.
• Annuities can be either inflows or outflows.
• An ord...
Annuities
Future Value of an Ordinary Annuity
Using the FVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How mu...
Future Value of an Ordinary Annuity
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will you...
Future Value of an Annuity Due
Using the FVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much wi...
Future Value of an Annuity Due
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your dep...
Present Value of an Ordinary Annuity
Using PVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much ...
Present Value of an Ordinary Annuity
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could y...
Present Value of an Annuity Due
Using PVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could...
Present Value of an Annuity Due
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you bo...
Present Value of a Perpetuity
• A perpetuity is a special kind of annuity.
• With a perpetuity, the periodic annuity or ca...
Future Value of a Mixed Stream
Using Tables
• A mixed stream of cash flows reflects no particular
pattern
• Find the futur...
Future Value of a Mixed Stream
Using EXCEL
• Find the present value of the following mixed stream
assuming a required retu...
Present Value of a Mixed Stream
Using Tables
• A mixed stream of cash flows reflects no particular
pattern
• Find the pres...
Present Value of a Mixed Stream
Using EXCEL
• Find the present value of the following mixed stream
assuming a required ret...
Compounding Interest More
Frequently Than Annually
• Interest is often compounded more frequently than
once a year. Saving...
Example:
Future Value from Investing P100 at 8% Interest Compounded
Semiannually over 24 Months (2 Years)
Period

Beginnin...
Example:
Future Value from Investing P100 at 8% Interest Compounded
Quarterly over 24 Months (2 Years)
Period

Beginning
P...
Example:
Future Value at the End of Years 1 and 2 from Investing P100 at
8% Interest, Given Various Compounding Periods
Co...
• FVIFi,n

= (1+i/m)mxn

• The basic equation for future value can
no w be rewritten as
FVIFi,n

= (1+i/m)mxn
USING COMPUTATIONAL TOOLS FOR
COMPOUNDING MORE FREQUENTLY
THAN ANNUALLY
• Semiannual
Quarterly
Input
100

Function

Input
...
Spreadsheet Use
A
1

B

FUTURE VALUE OF A SINGLE AMOUNT WITH SEMIANNUAL AND
QUARTERLY COMPOUNDING

2

Present value

3

In...
Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Thr...
Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Thr...
Continuous Compounding
• CALCULATOR USE
Input

Function

0.16

2nd

1.1735

100

x
=

Solution is 117.35
Continuous Compounding
• Spreadsheet Use
A
1

B

FUTURE VALUE OF SINGLE AMOUNT WITH
CONTINOUS COMPOUNDING

2

Present valu...
Nominal & Effective Rates
• The nominal interest rate is the stated or contractual
rate of interest charged by a lender or...
Nominal & Effective Rates
• For example, what is the effective rate of interest on
your credit card if the nominal rate is...
Special Applications of Time Value
Future value and present value techniques
have a number of important applications in
fi...
Determining Deposits Needed to
Accumulate a Future Sum
Supposed you want to buy a house 5 years from now,
and you estimate...
Determining Deposits Needed to
Accumulate a Future Sum
• Calculator Use
Input
3000

Function
FV

5

N

6

I

Solution
is
5...
Determining Deposits Needed to
Accumulate a Future Sum
Spreadsheet Use
A
1

B

ANNUAL DEPOSITS NEEDED TO ACCUMULATE A FUTU...
Loan Amortization
 The term loan amortization refers to the
determination of equal periodic loan
payments.
 Lenders use ...
Loan Amortization
You borrow P6000 at 10 percent and agree to
make equal annual end of year payments over
4 years.
PVAn = ...
Loan Amortization
• Calculator Use
Input

Function

6000

PV

4

N

10

I

Solution
is
1,892.82

CPT
PMT
Loan Amortization
Loan Amortization
A
1

B

ANNUAL PAYMENT TO REPAY A LOAN

2

Loan Principal (present value)

3

Annual rate of interest

4...
A
1
2

B
C
Loan Amortization
Data: Loan
Principal

D

E

P6000

3

Annual rate of interest

10%

4

Number of years

4

5
...
Loan Amortization
• Use Table A-4
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rat...
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rat...
Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rat...
Finding an unknown Number of
Periods
• Ann Bates wishes to determine the number of years it
will take for her initial P100...
Finding an Unknown Number
of Periods
A

B

1

YEARS FOR A PRESENT VALUE TO GROW TO A SPECIFIED
FUTURE VALUE

2

Present va...
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FINANCIAL MANAGEMENT PPT BY FINMAN Time value of money official

  1. 1. Principles of Managerial Finance Time Value of Money MBA 656 FINANCIAL MANAGEMENT CYCLE 1 BY: MARY ROSE HABAGAT GELITA COLON
  2. 2. WHY THIS TOPIC MATTERS TO YOU IN PROFESSIONAL LIFE: ACCOUNTING: You need to understand time-value-of-money calculations to account for certain transactions such as loan amortization, lease payments, and bond interest rates.
  3. 3. INFORMATION SYSTEM: You need to understand time-value-of-money calculations to design systems that accurately measure and value the firm’s cash flows. MANAGEMENT: You need to understand time-value-of-money calculations so that you can manage cash receipts and disbursements in a way that will enable the firm to receive the greatest value from its cash flows.
  4. 4. MARKETING You need to understand time value of money because funding for new programs and products must be justified financially using time-value-ofmoney techniques. OPERATIONS You need to understand time value of money because the value of investments in new equipment, in new processes, and in inventory will be affected by the time value of money.
  5. 5. IN YOUR PERSONAL LIFE Time value techniques are widely used in personal financial planning. You can use them to calculate the value of savings at given future dates and to estimate the amount you need now to accumulate a given amount at a future date. You also can apply them to value lump-sum amounts or streams of periodic cash flows and to the interest rate or amount of time needed to achieve a given financial goal.
  6. 6. Learning Objectives • Discuss the role of time value in finance and the use of computational aids used to simplify its application. • Understand the concept of future value, its calculation for a single amount, and the effects of compounding interest more frequently than annually. • Find the future value of an ordinary annuity and an annuity due and compare these two types of annuities. • Understand the concept of present value, its calculation for a single amount, and its relationship to future value.
  7. 7. Learning Objectives • Calculate the present value of a mixed stream of cash flows, an annuity, a mixed stream with an embedded annuity, and a perpetuity. • Describe the procedures involved in: – determining deposits to accumulate a future sum, – loan amortization, and – finding interest or growth rates
  8. 8. The Role of Time Value in Finance • Most financial decisions involve costs & benefits that are spread out over time. • Time value of money allows comparison of cash flows from different periods. Question? Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return $500,000 after two years?
  9. 9. The Role of Time Value in Finance • Most financial decisions involve costs & benefits that are spread out over time. • Time value of money allows comparison of cash flows from different periods. Answer! It depends on the interest rate!
  10. 10. Present Value and Future Value PRESENT VALUE FUTURE VALUE • Is the cash on hand today • • It is the amount you need today in to reach a future value • • PRESENT VALUE TECHNIQUE uses discounting to find its present valueof each cash flow at time zero and then sums these values to find the investment’s value today • Is cash you will receive at a given future date It is the amount you will receive in the future from your cash on hand FUTURE VALUE TECHNIQUE uses compounding to find future value of each cash flow at the end of the investment’s life and then sums these values to find the investment’s future value
  11. 11. ILLUSTRATION
  12. 12. Simple Interest With simple interest, you don’t earn interest on interest. • Year 1: 5% of $100 = $5 + $100 = $105 • Year 2: 5% of $100 = $5 + $105 = $110 • Year 3: 5% of $100 = $5 + $110 = $115 • Year 4: 5% of $100 = $5 + $115 = $120 • Year 5: 5% of $100 = $5 + $120 = $125
  13. 13. Compound Interest With compound interest, a depositor earns interest on interest! • Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00 • Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25 • Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76 • Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55 • Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63
  14. 14. Computational Aids • Use the Equations • Use the Financial Tables • Use Financial Calculators • Use Spreadsheets
  15. 15. Computational Aids Future value interest factor or present value interest factor
  16. 16. Computational Aids
  17. 17. Time Value Terms • PV0 = present value or beginning amount • k = interest rate • FVn = future value at end of “n” periods • n = number of compounding periods • A = an annuity (series of equal payments or receipts)
  18. 18. Four Basic Models • FVn = PV0(1+k)n = PV(FVIFk,n) • PV0 = FVn[1/(1+k)n] = FV(PVIFk,n) A (1+k)n - 1 k = A(FVIFAk,n) = A 1 - [1/(1+k)n] = A(PVIFAk,n) • FVAn = • PVA0 k
  19. 19. BASIC PATTERNS OF CASH FLOW • SINGLE AMOUNT: a lump sum amount either currently held or expected at some future date • ANNUITY: a level periodic stream of cash flow • MIXED STREAM: a stream of unequal cash flows that reflect no particular pattern
  20. 20. Future Value Example Algebraically and Using FVIF Tables You deposit $2,000 today at 6% interest. How much will you have in 5 years? $2,000 x (1.06)5 = $2,000 x FVIF6%,5 $2,000 x 1.3382 = $2,676.40
  21. 21. Future Value Example Using Excel You deposit $2,000 today at 6% interest. How much will you have in 5 years? PV k n FV? $ 2,000 6.00% 5 $2,676 Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5, , 2000)
  22. 22. Compounding More Frequently than Annually • Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. • As a result, the effective interest rate is greater than the nominal (annual) interest rate. • Furthermore, the effective rate of interest will increase the more frequently interest is compounded.
  23. 23. Compounding More Frequently than Annually • For example, what would be the difference in future value if I deposit $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, an (d) monthly? Annually: 100 x (1 + .12)5 = $176.23 Semiannually: 100 x (1 + .06)10 = $179.09 Quarterly: 100 x (1 + .03)20 = $180.61 Monthly: 100 x (1 + .01)60 = $181.67
  24. 24. Compounding More Frequently than Annually On Excel Annually PV $ Sem iAnnually Quarterly 100.00 k 12.0% n 5 FV $176.23 $ 100.00 0.06 10 $179.08 $ 100.00 Monthly $ 100.00 0.03 0.01 20 60 $180.61 $181.67
  25. 25. Continuous Compounding • With continuous compounding the number of compounding periods per year approaches infinity. • Through the use of calculus, the equation thus becomes: FVn (continuous compounding) = PV x (ekxn) where “e” has a value of 2.7183. • Continuing with the previous example, find the Future value of the $100 deposit after 5 years if interest is compounded continuously.
  26. 26. Continuous Compounding • With continuous compounding the number of compounding periods per year approaches infinity. • Through the use of calculus, the equation thus becomes: FVn (continuous compounding) = PV x (ekxn) where “e” has a value of 2.7183. FVn = 100 x (2.7183).12x5 = $182.22
  27. 27. Present Value Example Algebraically and Using PVIF Tables How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? $2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5 $2,000 x 0.74758 = $1,494.52
  28. 28. Present Value Example Using Excel How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit? FV k n PV? $ 2,000 6.00% 5 $1,495 Excel Function =PV (interest, periods, pmt, FV) =PV (.06, 5, , 2000)
  29. 29. Annuities • Annuities are equally-spaced cash flows of equal size. • Annuities can be either inflows or outflows. • An ordinary (deferred) annuity has cash flows that occur at the end of each period. • An annuity due has cash flows that occur at the beginning of each period. • An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.
  30. 30. Annuities
  31. 31. Future Value of an Ordinary Annuity Using the FVIFA Tables • Annuity = Equal Annual Series of Cash Flows • Example: How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years. FVA = 100(FVIFA,5%,3) = $315.25 Year 1 $100 deposited at end of year = $100.00 Year 2 $100 x .05 = $5.00 + $100 + $100 = $205.00 Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25
  32. 32. Future Value of an Ordinary Annuity Using Excel • Annuity = Equal Annual Series of Cash Flows • Example: How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years. PMT k n FV? $ 100 5.0% 3 $ 315.25 Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, )
  33. 33. Future Value of an Annuity Due Using the FVIFA Tables • Annuity = Equal Annual Series of Cash Flows • Example: How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years. FVA = 100(FVIFA,5%,3)(1+k) = $330.96 FVA = 100(3.152)(1.05) = $330.96
  34. 34. Future Value of an Annuity Due Using Excel • Annuity = Equal Annual Series of Cash Flows • Example: How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years. PMT $ 100.00 k 5.00% n 3 FV $315.25 FVA? $ 331.01 Excel Function =FV (interest, periods, pmt, PV) =FV (.06, 5,100, ) =315.25*(1.05)
  35. 35. Present Value of an Ordinary Annuity Using PVIFA Tables • Annuity = Equal Annual Series of Cash Flows • Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? PVA = 2,000(PVIFA,10%,3) = $4,973.70
  36. 36. Present Value of an Ordinary Annuity Using Excel • Annuity = Equal Annual Series of Cash Flows • Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? PMT I n PV? $ 2,000 10.0% 3 $4,973.70 Excel Function =PV (interest, periods, pmt, FV) =PV (.10, 3, 2000, )
  37. 37. Present Value of an Annuity Due Using PVIFA Tables • Annuity = Equal Annual Series of Cash Flows • Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? PVA = 2000(PVIFA,10%,3)(1+k) = $5,471.40 PVA = 2000(2.487)(1.1) = $5,471.40
  38. 38. Present Value of an Annuity Due Using Excel • Annuity = Equal Annual Series of Cash Flows • Example: How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest? PMT I n PV? $ 2,000 10.0% 3 $5,471.40 Excel Function =PV (interest, periods, pmt, FV) =PV (.10, 3, 2000, )
  39. 39. Present Value of a Perpetuity • A perpetuity is a special kind of annuity. • With a perpetuity, the periodic annuity or cash flow stream continues forever. PV = Annuity/k • For example, how much would I have to deposit today in order to withdraw $1,000 each year forever if I can earn 8% on my deposit? PV = $1,000/.08 = $12,500
  40. 40. Future Value of a Mixed Stream Using Tables • A mixed stream of cash flows reflects no particular pattern • Find the future value of the following mixed stream assuming a required return of 8%. Year Cashflow (1) No. of years Year Cash Flow (n) 9%,N(3) PVIF FVIF earning int. 1 (2) 400 0.917 Future Value [(1)x(3)] PV (4) $ 366.80 1 2 P11,500 800 = 4 5-1 2 14,0003 5-2 500 = 3 0.8421.360 673.60 P15,640 $ 0.7721.260 386.00 17,640 $ 3 12,9004 5-3 400 = 2 0.7081.166 283.20 15,041 $ 4 16,0005 5-4 300 = 1 0.6501.080 195.00 17,280 $ 5 18,000 5-5 = 0 PV 1.000 $1,904.60 Fixed value of mixed stream 18,000 P83,601.40
  41. 41. Future Value of a Mixed Stream Using EXCEL • Find the present value of the following mixed stream assuming a required return of 8%. A 1 B FUTURE VALUE OF A MIXED STREAM 2 Interest rate, pct/year 8% 3 Year Year-End Cash flow Excel Function Year Cash Flow 4 1 400 1 800 2 P11,500 P12,900 7 4 500 3 400 4 8 5 300 5 P18,000 5 2 6 3 9NPV Future $1,904.76Value P14,000 P16,000 P83,608.15 Entry in Cell B9 is =FV(B2,A8,0,NPV (B2,B4:B8)
  42. 42. Present Value of a Mixed Stream Using Tables • A mixed stream of cash flows reflects no particular pattern • Find the present value of the following mixed stream assuming a required return of 9%. Year Cash Flow PVIF9%,N PV 1 400 0.917 $ 366.80 2 800 0.842 $ 673.60 3 500 0.772 $ 386.00 4 400 0.708 $ 283.20 5 300 0.650 $ 195.00 PV $1,904.60
  43. 43. Present Value of a Mixed Stream Using EXCEL • Find the present value of the following mixed stream assuming a required return of 9%. A 1 2 3 4 5 6 7 8 9 B PRESENT VALUE OF A MIXED STREAM OF CASH FLOWS Interest rate, pct/year Year Cash Flow 1 2 3 4 5 9% Year Year-End Cash Flow 1 P400 2 P800 3 P500 4 P400 5 P300 400 800 500 400 300 NPVPresent Value $1,904.76 P1,904.76 Excel Function Entry in Cell B9 is =NPV(B2,B4:B8)
  44. 44. Compounding Interest More Frequently Than Annually • Interest is often compounded more frequently than once a year. Savings institutions compound interest semi-annually, quarterly, monthly, weekly, daily, or even continuously. SEMIANNUAL COMPOUNDING of interest involves two compounding periods within the year. Instead of the stated interest rate being paid once a year, onehalf of the stated interest rate is paid twice a year. QUARTERLY COMPOUNDING of interest involves four compounding periods within the year. One-fourth of the stated interest rate is paid four times a year.
  45. 45. Example: Future Value from Investing P100 at 8% Interest Compounded Semiannually over 24 Months (2 Years) Period Beginning Principal (1) Future Value interest factor (2) Future value at end of period [(1)x(2)] (3) 6 months P100.00 1.04 P104.00 12 months 104.00 1.04 108.16 18 months 108.16 1.04 112.49 24 months 112.49 1.04 116.99
  46. 46. Example: Future Value from Investing P100 at 8% Interest Compounded Quarterly over 24 Months (2 Years) Period Beginning Principal (1) Future Value interest factor (2) Future value at end of period [(1)x(2)] (3) 3 months P100.00 1.02 P102.00 6 months 102.00 1.02 104.04 9 months 104.04 1.02 106.12 12 months 106.12 1.02 108.24 15 months 108.24 1.02 110.41 18 months 110.40 1.02 112.62 21 months 112.61 1.02 114.87 24 months 114.86 1.02 117.17
  47. 47. Example: Future Value at the End of Years 1 and 2 from Investing P100 at 8% Interest, Given Various Compounding Periods Compounding Period End of Year Annual Semiannual Quarterly 1 P108.00 P108.16 P108.24 2 116.64 116.99 117.17 As shown, the more frequently interest is compounded, the greater the amount of money accumulated. This is true for any interest rate for any period of time.
  48. 48. • FVIFi,n = (1+i/m)mxn • The basic equation for future value can no w be rewritten as FVIFi,n = (1+i/m)mxn
  49. 49. USING COMPUTATIONAL TOOLS FOR COMPOUNDING MORE FREQUENTLY THAN ANNUALLY • Semiannual Quarterly Input 100 Function Input Function PV 100 PV 4 N 8 N 4 I 2 I Solution is 116.99 CPT FV Solution is 117.17 CPT FV
  50. 50. Spreadsheet Use A 1 B FUTURE VALUE OF A SINGLE AMOUNT WITH SEMIANNUAL AND QUARTERLY COMPOUNDING 2 Present value 3 Interest rate, pct per year compounded semiannually 4 Number of years 5 Future value with semiannual compounding 6 Present value 7 Interest rate, pct per year compounded quarterly 8 Number of years 9 Future value with quarterly compounding Entry in cell B5 is = FV(B3/2,B4*2,0) Entry in cell B9 is = FV(B7/4,B8*4,0,-B2,0) P100 8% 2 P116.99 P100 8% 2 P117.17
  51. 51. Continuous Compounding • With continuous compounding the number of compounding periods per year approaches infinity. • Through the use of calculus, the equation thus becomes: FVn (continuous compounding) = PV x (eixn) where “e” has a value of 2.7183. • Continuing with the previous example, To find the value at the end f 2 years of Fred Moreno’s P100 deposit in an account paying 8% annual interest compounded continuously
  52. 52. Continuous Compounding • With continuous compounding the number of compounding periods per year approaches infinity. • Through the use of calculus, the equation thus becomes: FVn (continuous compounding) = PV x (eixn) where “e” has a value of 2.7183.
  53. 53. Continuous Compounding • CALCULATOR USE Input Function 0.16 2nd 1.1735 100 x = Solution is 117.35
  54. 54. Continuous Compounding • Spreadsheet Use A 1 B FUTURE VALUE OF SINGLE AMOUNT WITH CONTINOUS COMPOUNDING 2 Present value P100 3 Annual rate of interest, compounded continously 8% 4 Number of years 2 5 Future value with continuous compounding P117.35 Entry in Cell B5 is =B2*EXP(B3*B4)
  55. 55. Nominal & Effective Rates • The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. • The effective interest rate is the rate actually paid or earned. • In general, the effective rate > nominal rate whenever compounding occurs more than once per year EAR = (1 + i/m) m -1
  56. 56. Nominal & Effective Rates • For example, what is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly? EAR = (1 + .18/12) 12 -1 EAR = 19.56%
  57. 57. Special Applications of Time Value Future value and present value techniques have a number of important applications in finance. We’ll study four of them in this section: 1.Determining deposits needed to accumulate a future sum. 2.Loan amortization 3.Finding interest or growth rates, and 4.Finding an unknown number of periods
  58. 58. Determining Deposits Needed to Accumulate a Future Sum Supposed you want to buy a house 5 years from now, and you estimate that an initial down payment of P30,000 will be required at that time. To accumulate the P30,000, you will wish to make equal annual end-ofyear deposits into an account paying annual interest of 6 percent. FVAn = PMT X (FVIFAi,n) PMT = FVAn FVIFAi,n FVIFAi,n) = 1x[ (1+i)n – 1] i
  59. 59. Determining Deposits Needed to Accumulate a Future Sum • Calculator Use Input 3000 Function FV 5 N 6 I Solution is 5,321.89 CPT PMT
  60. 60. Determining Deposits Needed to Accumulate a Future Sum Spreadsheet Use A 1 B ANNUAL DEPOSITS NEEDED TO ACCUMULATE A FUTURE SUM 2 Future value 3 Number of years 4 Annual rate of interest 5 Annual deposit Entry in Cell B5 is =-PMT(B4,B3,0,B2). Table Use: Use Table A-3 P30,000 5 6% P5,321.89
  61. 61. Loan Amortization  The term loan amortization refers to the determination of equal periodic loan payments.  Lenders use a loan amortization schedule to determine these payment amounts and the allocation of each payment to interest and principal.  Amortizing a loan actually involves creating an annuity out of a present amount.
  62. 62. Loan Amortization You borrow P6000 at 10 percent and agree to make equal annual end of year payments over 4 years. PVAn = PMT X (FVIFAi,n) PMT = PVAn PVIFAi,n PVIFAi,n = 1x[ 1 - 1 (1+i)n ]
  63. 63. Loan Amortization • Calculator Use Input Function 6000 PV 4 N 10 I Solution is 1,892.82 CPT PMT
  64. 64. Loan Amortization
  65. 65. Loan Amortization A 1 B ANNUAL PAYMENT TO REPAY A LOAN 2 Loan Principal (present value) 3 Annual rate of interest 4 Number of years 4 5 Annual payment P1,892.82 Entry cell B5 is = -PMT(B3,B4,B2) P6,000 10%
  66. 66. A 1 2 B C Loan Amortization Data: Loan Principal D E P6000 3 Annual rate of interest 10% 4 Number of years 4 5 Annual Payments 6 Year Total To interest To Principal Year-End Principal 7 0 8 1 1892.82 600.00 1,292.82 4,707.18 9 2 1892.82 470.72 1,422.11 3,285.07 10 3 1892.82 328.51 1,564.32 1,720.75 11 4 1892.82 172.07 1,720.75 0 6,000 Key Cell Entries Cell B8:=-PMT($D$3,$D$4,$D$2),copy t B9;B11 Cell C8:=-CUMIPMT($D$3,$D$4,$D$2,A8,A8,0), copy to C9:C11 CellD8:=-CUMPRINC($D$3,$D$4,$D$2,A8,A8,0),copy to D9:D11 Cell E8:=E7-D8,copy to E9:E11
  67. 67. Loan Amortization • Use Table A-4
  68. 68. Determining Interest or Growth Rates • At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows. • For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below? 1994 $ 1,000 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525 It is first important to note that although there are 7 years show, there are only 6 time periods between the initial deposit and the final value.
  69. 69. Determining Interest or Growth Rates • At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows. • For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below? 1994 $ 1,000 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525 PV FV n k? $ $ 1,000 5,525 6 33.0%
  70. 70. Determining Interest or Growth Rates • At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows. • For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below? 1994 $ 1,000 1995 1,127 1996 1,158 1997 2,345 1998 3,985 1999 4,677 2000 5,525 Excel Function =Rate(periods, pmt, PV, FV) =Rate(6, ,1000, 5525)
  71. 71. Finding an unknown Number of Periods • Ann Bates wishes to determine the number of years it will take for her initial P1000 deposit, earning 8% annual interest, to grow to equal P2,500. Simply stated, at an 8% annual rate of interest, how many years, n will it take for Ann’s P1000,PV, to grow to P2,500,FV? • Table Use: • We begin by dividing the amount deposited in the earliest year by the amount received in the latest year. This will result to present value interest factor • Use Table A-2
  72. 72. Finding an Unknown Number of Periods A B 1 YEARS FOR A PRESENT VALUE TO GROW TO A SPECIFIED FUTURE VALUE 2 Present value (deposit) 3 Annual Rate of Interest, compounded annually 4 Future value 2,500 5 Number of years 11.91 Entry in Cell B5 is =NPER(B3,0,B2,-B4). P1000 8%

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